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  • Music

    2021-07-06 08:57:10
    MusicConrad Prebys Music Centerhttp://musicweb.ucsd.eduAll courses, faculty listings, and curricular and degree requirements described herein are subject to change or deletion without notice.The Gradu...

    Music

    Conrad Prebys Music Center

    http://musicweb.ucsd.edu

    All courses, faculty listings, and curricular and degree requirements described herein are subject to change or deletion without notice.

    The Graduate Program

    UC San Diego offers the PhD with areas of emphasis in composition, computer music, and integrative studies, and the doctor of musical arts (DMA) in contemporary music performance. All applicants admitted to the graduate program will be officially entering the PhD or DMA program, with the prospect of completing a doctoral degree. Concurrent PhD or PhD and DMA degrees are not allowed. Applicants who have not previously earned a master’s degree in a music-related field from another institution will earn the MA while completing doctoral requirements. Students wishing to pursue a master’s degree only are encouraged to speak with the graduate adviser.

    Composition

    The Composition Program is committed to nourishing the individual gifts and capacities of student composers in a diverse and active environment, with an emphasis on intensive personal interaction between faculty and student. The faculty mentor considers a student’s particular goals and then attempts to strengthen his or her technical capacity to meet them. The diversity and liveliness of our program itself often challenges students to reevaluate their goals.

    An incoming member in the MA or PhD program begins with a yearlong seminar (taught by a different faculty composer each quarter) and continues with individual studies thereafter. At the close of the first-year fall quarter and again after the following spring quarter, the entire composition community gathers for a daylong “jury.” Each seminar member is allotted a block of time during which the composition that has just been completed is performed and recorded in a carefully rehearsed presentation. There is a detailed discussion of each work by the faculty composers, and the student has opportunity to comment, explain, and pose questions. Following the performance and discussions of this day, the composition faculty meets to assess the students’ work collectively and to offer any guidance deemed necessary. This process is at the root of the uniqueness of the UC San Diego program, and manifests the range, seriousness, and vitality with which compositional issues are explored here.

    After completing three quarters of seminar and two juries, students come to know something about the ideas and perspectives of each faculty composer; the faculty, in turn, is aware of each student’s objectives and needs. At this point, an individual mentor is agreed upon and this relationship becomes the center of the student’s continuing work as the degree is completed. A Third Year Forum presents, under departmental auspices, a work composed by each third-year PhD composer in the four quarters since his or her second jury. As a part of preparation for this forum, each student composer is expected to have a faculty performer on his or her PhD committee (as a regular member, or as an additional sixth member). The faculty performer is the student’s performance mentor and guide in interfacing with the performance community. There is also a biweekly Focus on Composition Seminar at which faculty, students, and selected visitors present work of interest (compositional, analytical, technological, and even whimsical).

    The seminars serve to foster mutual awareness within the student composer group. Collegial relationships develop and lead not only to friendships, but also to further creative outlets in cooperative projects, including the student-run Composers’ Forums, performance collectives, and recital projects. UC San Diego performers—faculty and student—are all committed to the playing of new music, and frequent composer/performer collaborations are a vital aspect of life in the Department of Music.

    Computer Music

    The Computer Music Program emphasizes research in new techniques for electronic music composition and performance, catalyzed through an active concert program of new works by students, faculty, and visitors. Areas of research may include

    new audio synthesis techniques

    audio signal processing

    music cognition

    live improvisation with and by computers

    integrating audio and video

    electronic spatialization of sounds

    techniques for live electronic music performance

    computer music software and HCL design

    audio analysis and feature detection

    The Computer Music Program encourages work that overlaps with the other programs of study: Composition, Performance, and Integrative Studies. Analyzing and performing electronic music repertoire as well as writing new music involving electronics are encouraged.

    The first-year computer music curriculum is centered on a yearlong “backbone” course covering the essentials of the computer music field. This material divides naturally into three portions (audio signal processing, compositional algorithms, and musical cognition).

    In their second year, students work individually with faculty members to deepen their mastery of their subject areas of concentration. For example, a student wishing to focus on signal processing aspects might study techniques for digital audio analysis and resynthesis, drawing on the current research literature.

    After having taken a critical mass of such subjects, PhD students enter a qualifying examination preparation period, and, once successful, they start their dissertation research.

    Integrative Studies (IS)

    (Formerly Critical Studies/Experimental Practices)

    Drawing on a wide range of academic fields, including critical and cultural theory, ethnomusicology, music cognition, new media studies, sound studies, and ecocriticism, among others, the IS program combines an exploration of contemporary music making with an examination of ideas and concepts that are relevant to its nature, creation, production, and reception. Exposure to a range of disciplines and interdisciplinary methods prepares students to pursue innovative scholarship and creative work.

    IS graduate students initially enroll in introductory courses taught by core faculty members designed to present intersecting ways of researching sound, music, and culture, and which are designed to generate possibilities for future independent and collaborative research. In subsequent quarters students choose between a variety of focused and revolving topic seminars. Recent seminars have included Sounding Sex, Race, and Gender; Post-Colonial Hermeneutics; Music, Sound, and Biopolitics; Reading Ethnomusicology; Theorizing Radio and Musical Identities; Critical Historiography; Music and Affect; Proseminar in Creative Practice; Scholarly Writing for Publication; and Arts of the Archive. Seminars offered in other departments—for instance, in visual arts, literature, theatre and dance, anthropology, communication, ethnic studies, cognitive science, psychology, or computer science—are encouraged and may fulfill degree requirements, if approved by a student’s faculty adviser.

    The integrative studies program embraces multiple ways of knowing and encourages cross-fertilization and hybridity between diverse disciplines and musical forms. Faculty and students in integrative studies produce work that moves fluidly between scholarship, performance, improvisation, sound installation, composition, instrument building, and more. The program teaches students to situate knowledge and practices from local to global communities, and to produce compelling scholarly writing and creative work that recognizes the responsibilities and opportunities associated with living in an increasingly interconnected and interdependent world.

    Performance

    Fostering the creative, intelligent, and passionate performance of contemporary music is the mission of the Performance Program of the Department of Music. As once stated by founding faculty composer Robert Erickson, we at UC San Diego are a “community of musicians.”

    Performers act and interact in a communal environment by means of collaboration with faculty and student composers, research in the areas of new performance modalities, music technology, and improvisation, among many other pursuits. The performance of contemporary music is viewed as a creative act that balances expertise and exploration.

    Graduate performance students pursue either a master of arts or a doctor of musical arts degree in contemporary music performance. The course of study for both programs involves the completion of required graduate seminars and intensive study with a mentoring faculty member. Students are encouraged to adopt a vigorous, exploratory orientation in their private study. Final degree requirements include a recital, or in the case of the DMA, two recitals and the presentation of personal performance research.

    The work of graduate performance students forms an integral component of a rich musical environment, which produces an astonishing quantity and variety of performances. Students may perform in collaborative performances with fellow students and faculty. Ensembles include groups specializing in the interpretation of unconventionally notated scores, the percussion group red fish blue fish and other ensembles. The Performance Forum, a student-initiated concert series, provides an opportunity for students to present a wide variety of repertoire that may include improvised music, world music, and music with technology. A strong, collaborative spirit among the curricular areas of the department (Performance, Composition, Music Technology, and Integrative Studies) also yields many new projects each year. Works by graduate student composers are performed on the annual Spring Festival and other concert series. The sense of musical community engendered by diverse interactions permeates the atmosphere and makes the Department of Music at UC San Diego a uniquely rewarding place to create the newest of music.

    Graduate Admissions

    Students are admitted to begin in fall quarter only. The deadline for submission of ALL application materials is December 4. Failure to meet this deadline jeopardizes admission and financial support. The admissions procedure is handled through Grad Apply beginning on September 1. The following PDF documents must be submitted to Grad Apply: statement of purpose; three letters of recommendation; a minimum of two papers illustrating writing ability in any area of music scholarship related to your degree goals (Your most recent writing and writing on twentieth- and twenty-first-century musical practices is preferred.); additional documentation of previous work (See below for more information from each area of emphasis.); and unofficial transcripts from all institutions attended (Scanned copies are acceptable. Graduate Record Examination (GRE) scores are not required for this application. Upon provisional admission to graduate study at UC San Diego, official hardcopy transcripts are required to finalize your admission). For international students: The TOEFL exam (Test of English as a Foreign Language) or IELTS (International English Language Testing System) is required of ALL international students whose country of citizenship does not have English as its primary language, unless you have been enrolled in a full-time program of study for a minimum of one year in a country whose primary language is English. The minimum TOEFL score is 550 (or 213 for the computer-based exam). The minimum IELTS score is 7. The minimum speaking TOEFL score is 25 and 28–30 for IELTS.

    Materials to Submit for Area of Emphasis

    Composition applicants: Three scores of instrumental works,ideally accompanied by recorded examples of the works being performed. Such evidence may include, but should not be exclusively, electroacoustic works or installation documentation. At least one of the submitted works must be a score involving traditional instrumental writing. A repertory listof works (solo and chamber) performed or composed during the past few years and digital or scanned samples of printed concert programs in which you have participated, either as performer, composer, or collaborator.

    Computer music applicants: Appropriate documentation of prior work (e.g., papers, performances, intermedia works, computer programs, etc.).

    Integrative studies applicants: At least one of your writing samples should represent substantial and sustained engagementwith a topic of relevance to contemporary music scholarship. If appropriate, documentation of performances, compositions, computer music projects, installations, or intermedia work.

    Performance applicants: Audio filesdemonstrating the level of vocal and/or instrumental performance. In-person auditions are desirable when possible, but not required. A repertory listof works (solo and chamber) performed or composed during the past few years and digital/scanned samples of printed concert programs in which you have participated, either as performer, composer, or collaborator.

    Core Graduate Curriculum

    Methods:All students are required to complete both courses during their first quarter at UC San Diego.

    MUS 291. Introduction to Research Methods

    MUS 501. Introduction to Teaching Methods

    Performance:All students must complete at least eight units.

    MUS 200, MUS 201, MUS 202

    Depth: All students must complete at least thirty-two units from among these options.

    MUS 206

    MUS 207

    MUS 210

    MUS 215A-B-C (IS area requirement; see note below)

    MUS 267

    MUS 270A-B-C (computer music area requirement; see note below)

    MUS 271A-B-C

    MUS 272

    MUS 170–177 (No more than one course from this series can count toward this requirement.)

    Note: Courses may not be used to satisfy both area requirements and methods/depths units unless specifically stated.

    All courses in the methods and depth series above must be taken for a letter grade to count toward a student’s degree progress.

    Breadth:All students are encouraged to take at least one graduate-level or upper-division undergraduate course outside of the department, which, by petition and on a case-by-case basis, may count toward the depth requirement.

    Focus:All students (except for computer music) are required to enroll in the appropriate area focus course (S/U grading option only) every quarter in residence (for PhD students), or until advanced to candidacy (for DMA students).

    MUS 204, MUS 205, MUS 245

    Research: All students must complete at least six units of MUS 299 and are encouraged to pursue independent research on a continuing basis. Doctoral students must complete at least six units of MUS 298 enrolled with members of the student’s doctoral committee in preparation for the qualifying exam.

    MUS 298, MUS 299

    Teaching: Participation in the undergraduate teaching program is required of all graduate students at the equivalent of 25 percent time for three quarters or 33 percent for two quarters (six units total).

    MUS 500

    Engagement:All students are encouraged to explore outreach and service opportunities during their graduate study and to engage in sustained and substantive ways with our diverse local communities as an integral part of their creative and scholarly research.

    Note: All courses in the methods and depth series above must be taken for a letter grade to count toward a student’s degree progress.

    Area Requirements

    In addition to the core graduate and PhD or DMA curriculum, doctoral students (according to their area of emphasis) must complete the following courses prior to the qualifying examination:

    Composition

    MUS 203A-B-C—successful completion of the jury process is necessary to obtain a passing grade in the corresponding seminar.

    MUS 203D—every quarter in residence with committee chair after successful completion of 203C.

    MUS 210—must be taken twice (one time can count toward the depth core requirement).

    MUS 228

    MUS 229

    MUS 271A-B—both courses can count toward “depth” requirement.

    Courses may not be used to satisfy both area requirements and methods/depths units unless specifically stated.

    Computer Music

    MUS 270A-B-C—must be taken within the first year of the program unless previously taken as a UC San Diego MA student.

    MUS 270D—required to be taken a total of three times.

    Courses may not be used to satisfy both area requirements and methods/depths units unless specifically stated.

    Integrative Studies

    MUS 215A-B-C

    MUS 215D—must be taken twice after successful completion of MUS 215C, prior to qualification.

    Courses may not be used to satisfy both area requirements and methods/depths units unless specifically stated.

    Performance

    MUS 200, 201, or 202—every quarter in residence.

    MUS 232—every quarter in residence.

    Courses may not be used to satisfy both area requirements and methods/depths units unless specifically stated.

    Preliminary Examination

    Each graduate program has area-specific requirements constituting a “preliminary exam” that takes place during fall of the student’s second year. The purpose of the preliminary examination is to evaluate a student’s potential to succeed in the program and their command of content presented in the first year of course work. If the participating professors unanimously agree that the student has not passed the exam, then the student will be allowed to finish the second year and to submit MA completion requirements but will not be allowed to continue with the doctoral program. The overriding purpose of the exam, however, is constructive rather than punitive.

    Composition—The student’s second jury piece will be evaluated.

    Computer Music—Students submit one research paper for evaluation.

    Integrative Studies—Students complete an exam consisting of written responses to questions provided by the professors who taught in the core MUS 215A-B-C sequence.

    Performance—Students are asked to give a presentation showcasing and/or discussing portions of the material planned for their first recital.

    Master’s Degree Completion Requirements

    If a student opts for the terminal MA or wishes to earn an MA along the way to the doctorate, they would fulfill these completion requirements. However, students are not required to complete the MA en route to the doctorate.

    Successful completion of the core curriculum to the MA level and all curricular requirements of the student’s area of emphasis.

    Formation of a master’s committee (see UC San Diego policy on appointment of the master’s committee).

    Submission of a substantive and polished research paper (which may have originated as a graduate seminar paper) to the student’s MA committee for approval.

    Students emphasizing composition must submit a folio of three chamber compositions (supplemented by audio or audio-visual recordings for at least two).

    Students emphasizing computer music must submit a second substantive and polished research paper along with either a thesis demonstrating original research, or a lecture-performance in which the scientific, technological, and musical aspects of an original computer music composition are documented, played, and discussed.

    Students emphasizing integrative studies must complete a thesis of roughly sixty pages demonstrating original research and critical insight, which may also include documentation of their creative practice.

    Students emphasizing performance must present a major recital that merits approval by the student’s committee chair and submit either two additional creative projects with accompanying documentation or one creative project plus a second research paper.

    All of the above master’s requirements must have final approval from all members of a student’s committee.

    Qualifying Examination/Advancement to Candidacy

    All students are expected to advance to candidacy (i.e., qualify) by the end of the student’s third year of graduate study.

    Requirements prior to taking the qualifying examination:

    Successful completion of the core curriculum and all curricular requirements of the student’s area of emphasis for the PhD or DMA degree, including at least six units of MUS 298. Students should enroll in MUS 298 with the committee members who will be providing qualifying questions for the exam.

    Appointment of the doctoral committee per UC San Diego Senate regulations.

    All PhD students must submit one research paper (not previously submitted for any other degree) that is judged by the student’s committee to be of publishable quality. The topic and scope of the publishable paper will be developed with, and approved by, the student’s PhD committee chair.

    Composition students must submit a folio of not fewer than three compositions.

    DMA students must give at least one recital.

    The qualifying examination for all doctoral students will consist of the following:

    A written and oral defense of three questions provided by the doctoral committee pertaining to appropriate areas of specialization. Qualifying exams must be well supported with appropriate and proper citations and are most often in the range of twenty-five pages of written text for each question. Students have a twelve-day period in which to write the qualifying exam and must deliver copies of their responses to each committee member and to the graduate staff adviser within that twelve-day period. Final deliveries must be made on a regular UC San Diego workday, followed by a normative period of at least five workdays before the exam defense can be held, so please schedule accordingly. Students are advised to schedule an agreeable date and time for the exam with their committee members and then work backwards to arrive at an acceptable date for receiving their questions. For integrative studies students, one question will involve a defense of the student’s dissertation prospectus. This prospectus is a document that presents the research topic of the dissertation, places it in the context of relevant literature and/or in the context of recent artistic developments, discusses its significance, specifies and justifies the research methods, theoretical orientation and/or artistic approach, and indicates the anticipated steps leading to completion.

    Students are expected to advance to doctoral candidacy within their first three years. If final approval is not obtained, students will either not be allowed to continue in the program or will be placed in a one-quarter probationary period and asked to redo aspects of the qualifying process. If a student fails to gain final approval during this period, they will not be allowed to continue in the program and will receive no further funding or support.

    Successful completion of the qualifying exam marks the student’s advancement to doctoral candidacy.

    PhD and DMA Completion Requirements

    After successful completion of the qualifying examination, all students must remain in residence for at least three quarters, during which time they must enroll in twelve units of MUS 299 every quarter with their committee chair or other committee members. Students must provide a full copy of the doctoral research they wish to defend to the student’s doctoral committee members four weeks prior to the doctoral defense. Materials previously submitted for other degrees are not acceptable, and in all cases a final public defense of the student’s doctoral work is required. It is understood that the edition of the dissertation given to committee members will not be the final form, and that committee members may request changes or revisions be made to the text after the defense. In extreme cases, another public defense may be merited.

    For compositionstudents, completion of a major composition project.

    For computer musicstudents, completion of an acceptable dissertation.

    For integrative studiesstudents, completion of a book-length dissertation demonstrating original research and critical insight, or presentation of a major creative work and a substantive written defense of that work.

    For DMA students, completion of two more recitals, or one recital plus one of the following: (a) a thesis or research project; (b) a concert or lecture recital that is innovative in design and/or content and which is supported by appropriate documentation as determined by the committee; or (c) two approved chamber music concerts, with appropriate documentation as determined by the committee.

    All of the above doctoral completion requirements must have final approval from all members of a student’s committee. Acceptance of the dissertation by the university librarian represents the final step in completion of all requirements for the PhD.

    Materials previously submitted for degrees at other institutions are not acceptable for submission at UC San Diego.

    Specialization in Critical Gender Studies

    Students in the doctoral (PhD) program in music may apply for a specialization in critical gender studies to complement their course work and research in music.

    The Critical Gender Studies Program is built on the intellectual foundations of intersectional feminist thought and queer studies, and incorporates the interdisciplinary methodologies, intersectional frameworks, and transformational epistemologies central to contemporary gender and sexuality studies. The graduate specialization in critical gender studies provides specialized training in gender and sexuality for students currently enrolled in a UC San Diego doctoral (PhD) program. Through advanced course work in critical gender studies and its affiliated departments, graduate students in the specialization develop an understanding of gender as necessarily linked to other social formations, including sexuality, race, nation, religion, (dis)ability, and structures of capital. At the same time, doctoral (PhD) students engaging gender and sexuality studies have the opportunity to develop their work among peers who take up similar questions in their scholarship.

    Admitted students are required to complete five courses in addition to their home department’s core requirements, consisting of two core courses and three electives. The core courses are Advanced Studies in Critical Gender Studies (CGS 200), to be taken shortly after admission to the specialization, and Practicum in Critical Gender Studies (CGS 299), to be taken in the student’s final two years of dissertation writing. Electives may be chosen from a list of preapproved seminars in participating departments (students may petition other courses with significant gender/sexuality studies content) and may be taken at any time during the student’s tenure at UC San Diego. Admitted students must also include at least one member of their dissertation committee from the list of CGS core or affiliate faculty.

    For more information about the graduate specialization in critical gender studies, please visit

    Normative Support, Residency, and Time to Degree Policy

    California Residency Expectation

    Nonresident US students are expected to gain California residency after their first year. Continuing financial support is dependent on meeting this expectation.

    Departmental Support Policy

    All admitted graduate students are guaranteed support for up to fifteen quarters (five academic years). Financial support is contingent upon full-time registration (twelve units or more per quarter), making satisfactory progress toward degree completion, and being in good academic and employment standing. A typical funding package consists of tuition, health insurance, and student fees, plus a combination of student employment and/or stipend that equates to the salary of a 50 percent TA or GSR position over nine months. Students may apply for research travel support and summer teaching opportunities (especially after passing the qualifying exam). Employment during summer session does not count toward quarter limits, but employment for the colleges or elsewhere on campus during the academic year does count toward the above department support limits.

    Time to Degree Policy

    All applicants admitted to the graduate program will be officially entering the PhD or DMA program with the prospect of completing a doctoral degree.

    Each graduate program has area-specific requirements constituting a “preliminary exam” that takes place during fall of the student’s second year. The purpose of the preliminary examination is to evaluate a student’s potential to succeed in the program and their command of content presented in the first year of course work. If the participating professors unanimously agree that the student has not passed the exam, then the student will be allowed to finish the second year and to submit MA completion requirements but will not be allowed to continue with the doctoral program.

    All students are expected to advance to candidacy (i.e., qualify) by the end of their third year of graduate study.

    Departmental funding is dependent on meeting all of the above expectations. Students who do not comply with the above policy may be ineligible for their final three quarters of support.

    If final approval from all members of a student’s qualifying or dissertation committee is not obtained, students will either not be allowed to continue in the program or will be placed in a one-quarter probationary period and asked to redo aspects of the completion requirements. If a student fails to gain final approval during this probationary period, they will not be allowed to continue in the program and will receive no further funding or support.

    UC San Diego Time to Doctorate Policy

    The normal period in which doctoral students, under usual circumstances, are expected to complete the requirements for the degree is five years. In addition, the Department of Music Time to Doctorate policy includes maximum registered time in which a student must advance to doctoral candidacy (four years), maximum time during which a doctoral student is eligible for support (six years), and maximum registered time in which a student must complete all doctoral requirements (six years).

    Advising Office

    Graduate Staff Adviser

    Dimple Bhatt

    Room 197, Conrad Prebys Music Center

    (858) 534-3279

    Email: mus-grad@ucsd.edu

    展开全文
  • 1.$MUSIC$算法简介$MUSIC$算法由R.O.Schmidt在1979年提出,该算法利用了信号的频率向量和噪声子空间的正交性构造空间谱函数,通过谱峰搜索来估计信号频率。$MUSIC$算法需要构造扫描函数如下:$$\hat{P}_{MUSIC}\left...

    1.$MUSIC$算法简介

    $MUSIC$算法由R.O.Schmidt在1979年提出,该算法利用了信号的频率向量和噪声子空间的正交性构造空间谱函数,通过谱峰搜索来估计信号频率。

    $MUSIC$算法需要构造扫描函数如下:$$\hat{P}_{MUSIC}\left( w \right ) = \frac{1}{a^{H}\left(w \right)GG^{H}a\left( w \right )} = \frac{1}{\sum_{i=K+1}^{M}\left |a^{H}\left( w \right )u_{i} \right |^{2}},\quad w\epsilon \left [ -\pi,\pi \right ]$$

    上式中,$M$代表信号自相关矩阵的阶数,$K$代表频率源个数,$a_{w}=\begin{bmatrix}1\\e^{-jw}\\ \vdots \\e^{-j\left(M-1 \right )w}

    \end{bmatrix}$是信号频率向量;$u_{i}$是信号自相关矩阵的特征向量,可以看到我们只取后$\left( M-K \right)$个特征值对应的特征向量,即最小的(M-K)个特征值对应的特征向量(噪声子空间)。

    $MUSIC$算法就是在扫描$w$,理想情况下$w$扫描到信号频率点时$a^{H}\left(w \right)GG^{H}a\left( w \right )=0$。

    在实际工程中,由于使用信号样本的时间自相关$\hat{R}$代替$R$进行特征分解,所以$a^{H}\left(w \right)GG^{H}a\left( w \right )$在信号频率点处并不严格为0,而是一个很小的值,于是$\hat{P}_{MUSIC}\left( w \right )$在该频率点的值较大。

    所以$\hat{P}_{MUSIC}\left( w \right )$的峰值的位置反映了信号的频率值,但$\hat{P}_{MUSIC}\left( w \right )$并不是信号的功率谱,所以通常将它称之为伪谱或$MUSIC$谱。

    1.1 $MUSIC$算法步骤根据观测样本值$x\left( 0 \right),x\left ( 1 \right),\cdots,x\left( N-1 \right)$,估计自相关矩阵$\hat{R}\epsilon C^{M\times{M}}$。

    对$\hat{R}$进行特征分解,得到$M-K$个最小特征值对应的特征向量,构造矩阵$G$。

    在$\left[ -\pi,\pi \right]$内改变$w$,计算$\hat{P}_{MUSIC}\left( w \right )$,峰值位置就是信号角频率的估计值。

    2.$ROOT-MUSIC$算法

    $MUSIC$算法通过搜索$w$来估计信号频率,实质上是做了一个频率的遍历,$ROOT-MUSIC$算法将$MUSIC$算法$a_{w}=\begin{bmatrix}1\\e^{-jw}\\ \vdots \\e^{-j\left(M-1 \right )w}\end{bmatrix}$中的$e^{jw}$看作复数$z$,则可以得到:$$a^{H}\left(w \right)GG^{H}a\left( w \right )=a^{H}\left(z \right)GG^{H}a\left( z \right )=0\quad(1)$$

    所以,$z_{k}=e^{jw_{k}},k=1,\cdots,K$(信号的$K$个频率点),就是方程(1)的根,信号频率的估计由搜索/遍历问题转化成了一元高次方程的求根问题,这也是该算法被叫做$ROOT-MUSIC$的原因。

    将方程(1)做变换以后可以得到修正后的方程如下:$$a^{T}\left(z^{-1} \right)GG^{H}a\left( z \right )=0\quad(2)$$

    方程(2)共有$2\left( M-1 \right)$个根,但只有位于单位圆上的$K$个根才是需要的解。

    在实际应用中,使用有限时间样本信号的时间自相关矩阵$\hat{R}$代替随机信号的统计自相关矩阵$R$,所以求解方程(2)得到的$K$个根并不准确位于单位圆上,而是位于单位圆附近,所以求解方程后应该找最接近单位圆的$K$个根,这些根就是信号频率的估计。

    2.1 $ROOT-MUSIC$算法步骤根据观测样本值$x\left( 0 \right),x\left ( 1 \right),\cdots,x\left( N-1 \right)$,估计自相关矩阵$\hat{R}\epsilon C^{M\times{M}}$。

    对$\hat{R}$进行特征分解,得到$M-K$个最小特征值对应的特征向量,构造矩阵$G$。

    求解方程(2),找出其中最接近单位圆的K个根,这些根的相位就是信号频率的估计。

    3.算例

    设随机过程$u \left( n \right)$为$u \left( n \right) = e^{j0.5\pi n + j \phi {1}} + e^{-j0.3\pi n + j \phi {2}} + v_{n}$,其中,$v_{n}$是0均值,方差为1的白噪声,$\phi_{1}$、$\phi_{2}$是相互独立并在$\left[ 0,2\pi \right]$上服从均匀分布的随机相位,使用$MUSIC$算法和$ROOT-MUSIC$算法进行信号频率估计。

    4.$Matlab$仿真结果1MUSIC算法频率估计结果7464a1cc18b7a7ae62e84772fb972c45.pngROOT-MUSIC算法结果

    5.$Matlab$代码实现

    $MUSIC$算法和$ROOT-MUSCI$算法具体实现如下,点击下载代码!N=1000;%信号样本数

    noise=(randn(1,N)+1j*randn(1,N))/sqrt(2);%产生噪声

    %产生信号

    c1=2*pi*rand;

    c2=2*pi*rand;

    signal1=exp(1j*(0.5*pi*(0:N-1)+c1));

    signal2=exp(1j*(-0.3*pi*(0:N-1)+c2));

    %观察样本

    s=signal1+signal2+noise;

    M=8;%自相关矩阵的阶数

    for i=1:N-M

    xx(:,i)=s(i+M-1:-1:i).'; %构造样本矩阵

    end

    R=xx*xx'/(N-M);%自相关矩阵

    [EV,D]=eig(R);%特征值分解

    EVA=diag(D)';

    [EVA,I]=sort(EVA);%特征值从小到大排序

    EVA=fliplr(EVA);%左右翻转,从大到小排序

    EV=fliplr(EV(:,I));%对应特征矢量排列

    G=EV(:,3:M); %噪声子空间

    NF=2048;

    %MUSIC算法

    w=linspace(-pi,pi,NF);

    for ii=1:NF

    a=exp(-1j*w(ii)*(0:M-1)');

    Pmusic(ii)=1/(a'*G*G'*a);

    end

    Pmusic=abs(Pmusic)/max(abs(Pmusic));

    plot(w/2/pi,10*log10(Pmusic));

    xlabel('w/2/pi')

    ylabel('归一化功率谱 (dB)')

    title('MUSIC算法');

    %root—music算法

    GG=G*G';

    co=zeros(2*M-1,1);%初始化3.6.38的2*(M-1)次方程的系数

    for m=1:M

    co(m:m+M-1)=co(m:m+M-1)+GG(M:-1:1,m);%计算3.6.38左边的多项式系数

    end

    z=roots(co);%多项式求根

    ph=angle(z)/(2*pi);%归一化频率

    err=abs(abs(z)-1);%求2(M-1)个根与单位圆之间的距离

    [err1,I]=sort(err);%将距离误差从小到大排序构成一个列向量

    f=[ph(I(1)),ph(I(3))];%选择误差最小的二个值所对应的归一化频率

    zz=[z(I(1)),z(I(3))];

    ff=sort(f);

    展开全文
  • MUSIC算法对信号DOA的应用波达方向(DOA)估计的基本问题就是确定同时处在空间某一区域内多个感兴趣的信号的空间位置(即多个信号到达阵列参考阵元的方向角)。最早的也是最经典的超分辨DOA估计方法是著名的MUSIC方法,...

    MUSIC算法对信号DOA的应用

    波达方向(DOA)估计的基本问题就是确定同时处在空间某一区域内多个感兴趣的信号的空间位置(即多个信号到达阵列参考阵元的方向角)。最早的也是最经典的超分辨DOA估计方法是著名的MUSIC方法,MUSIC是多重信号分类(Multiple Signal Classification)的英文缩写。它是由R.O. Schmidt于1979年提出来的,由1986年重新发表的。MUSIC算法利用了信号子空间和噪声子空间的正交性,构造空间谱函数,通过谱峰搜索,检测信号的DOA.它是建立在以下假设基础上的:

    (1) 阵列形式为线性均匀阵,阵元间距不大于处理最高频率信号波长的二分之一;

    (2) 处理器的噪声为加性高斯分布,不同阵元间距噪声均为平稳随机过程,独立同分布,空间平稳(各阵元噪声方差相等);

    (3) 空间信号为零均值平稳随机过程,它与阵元噪声相互独立;

    (4) 信号源数小于阵列元数,信号取样数大于阵列元数,信号源为窄带信号,即信号通过天线阵的时间远远小于信号带宽的倒数.

    5.2.1 MUSIC算法的基本原理

    图5.1 均匀天线阵列

    如图5.1,M个天线阵元均匀直线排列,单元间距d为1/2个波长,布置成一个阵列天线。设有P(P

    X(n)=AS(n)+U(n)   n=1,2,……N    (5.1)

    式中X(n)= 为M个阵元输出;

    A= ,

    式中  ,T表示转置, 为载波波长,i=1,2,……,P; 为第i个平面波的复振幅;U(n)= , 为零均值、方差为 的白噪声,且与信号源不相关;N为采样数。

    信号和噪声的协方差矩阵分别为

    S= U=

    接收信号的协方差(阵列输出信号协方差)

    ,以上式中H为共轭转置   (5.2)

    因为 为MXM矩阵,所以能分解为M个特征值和特征向量,把这些特征值和特征向量用 , (i=l,2,…,M)来表示,则 可表示为

    (5.3)

    这里,V是以 为元素的列矩阵, 是以 为元素的对角矩阵。从这个分析结果,有下面重要性质:

    [性质1] 各到达波是非相干(信号间相关系数不到l),设各信号和噪声不相关,在 的特征值里,下面关系成立

    (5.4)

    即主要的特征值(信号特征值)个数和到达波束P相等,剩下的特征值(噪声特征值)的大小等于噪声功率。根据这个性质可以估计到达波的个数。进一步,按照特征值分布, 可分为信号功率和噪声功率之和

    = =  (5.5)

    V=[ ]=

    由于特征向量相互正交,则由下面第二个重要的性质。

    [性质2] 对应噪声特征值的特征向量(噪声特征向量)和各到达波的信号向量(信号特征向量)正交 。

    …M, i=1,…P.    (5.6)

    于是,阵列的空间谱函数可表示为

    (5.7)

    式中分母是信号向量和噪声向量的内积。在性质2成立时的 分母是零, 有一尖峰。MUSIC算法就是通过寻找波峰来估计到达角的。通常把信号特征矢量覆盖的空间称为信号子空间(Signal Subspace),噪声特征向量覆盖的空间称为噪声子空间(Noise Subspace)。把基于这个原理的估计到达波方向的方法称为部分空间法(Subspace Method)。MUSIC算法就是用信号或噪声子空间进行低秩信息的提取。

    5.2.2 MUSIC算法的实现

    MUSIC算法的实现步骤:

    1) 根据N个接收信号矢量得到阵列输出向量的协方差矩阵

    (5.8)

    对上面的协方差矩阵进行特征值分解

    (5.9)

    2) 然后按特征值的大小顺序,把与信号个数P相等的特征值和对应的特征向量看作信号子空间,把剩下的(M-P)个特征值和特征向量看作噪声部分空间。

    = (5.10)

    3) 使 变化,按照空间谱 来计算谱函数,通过寻找峰值来得到信号到达方向的估计值。

    以下给出基于MATLAB的MUSIC算法估计仿真:

    (1)从 入射的三个独立信号源,SNR分别为12dB,10dB,9dB。

    图5.2 MUSIC算法的谱图

    从谱图可以看出:在满足上面的假设前提下,MUSIC算法可精确估计出信号的DOA。

    尽管MUSIC算法在满足上述假设前提下可以精确估计信号的DOA,但它也有局限性:就是在低SNR和小样本的条件下无法分辨出空间相距比较近的信号。还有就是在现实当中,由于多径效应,接收到的信号一般是高相关信号,甚至是相关信号。当阵列接收到的是相干信号时,MUSIC算法就失去了其有效性,不再能估计出信号的DOA了 。

    (2)如下图,从 入射的三个信号源,SNR分别是20dB、10dB、12dB,其中,后面两个是相干信号。

    图5.3 相干源的MUSIC谱图

    由上面的谱图可以看出:MUSIC算法无法分辨出 信号,即MUSIC算法对于相干信号的DOA估计完全失效。

    (3)如下图,三个分别从 入射的信号源,SNR分别为8dB,6dB,5dB。

    图5.4 相隔比较近的小信噪比信号的MUSIC谱图

    由谱图可以看出:MUSIC算法无法分辨 和 这两个信号,即MUSIC算法对于相隔比较近的小信噪比信号的DOA估计已经失效。

    针对上述情况,就必须找到一种新的算法或对MUSIC算法进行改进,使它在能区分一般环境下信号的基础上,也能分辨出相干信号的DOA和相隔比较近的小信号比信号的DOA。下面讨论一种修正的MUSIC算法。

    5.3修正MUSIC算法对信号DOA的估计

    MUSIC算法实现对信号源DOA的估计,是基于对阵列输出信号协方差进行特征分解来估计来波方向的。然而,若信号源中有某些源是相关或完全相关(相干),相干的几个信号就可能合并成一个信号,到达阵列的独立源数将减少,即阵列输出信号协方差的秩rank( )<P,对信号协方差矩阵进行特征值分解后,某些相干源的方向矢量不正交与噪声子空间,不出现信号零点。所以,有些源在空间谱曲线中将不呈现峰值,造成谱估计的漏报。

    对于小信噪比以及角度相隔比较近的信号,它们的阵列信号协方差矩阵进行特征值分解后同样会出现类似的情况,从而不能准确地估计信号的DOA。

    因此要对MUSIC算法进行改进,就是要对阵列输出信号协方差矩阵进行处理,使信号协方差的秩恢复为rank( )=P,从而能有效地估计出信号的DOA。

    空间平滑法较好地解决了相干信号源的情况,但它是以牺牲天线的有效阵元数为条件的,同时也增加了计算量。同时它对小信噪比信号和到达角度相隔比较近的信号不能分辨。本节研究的是一种修正的MUSIC算法,它在实现MUSIC算法功能的基础上,能分辨出上述三种环境下的信号。

    5.3.1 修正MUSIC算法的基本原理

    阵列输出信号的协方差矩阵为

    其中   X(n)=AS(n)+U(n) n=1,2,……N       (5.11)

    式中X(n)= 为M个阵元输出;A= ,        , ,T表转置, 为载波波长,i=1,2,…,P; 为第i个平面波的复振幅;U(n)= , 为零均值、方差为 的白噪声,且与信号源不相关;N为采样数。

    令I为MxM反向单位矩阵,即

    I=

    构造 RXX     5.12)

    这样做是使RXX成为Hermite的Toeplitz矩阵。Toeplitz矩阵的任何一条对角线取相同元素,关于副对角线对称的。由于协方差矩阵 是Hermite的Toeplitz矩阵,所以满足 = 。阵列输出矢量N次采样数据组成矩阵X=[X( ),X( ),…X( )],协方差矩阵的估值为 。一般情况下 只是Hermite矩阵,不是Toeplitz矩阵,利用 是Toeplitz性质对 进行修正,得到Toeplitz的协方差矩阵的估值RXX= + ,显然RXX是Hermite的Toeplitz矩阵,由此可知,RXX是 的无偏估计。再对RXX进行奇异值分解 ,有

    [U,S,V]=svd(RXX) (5.13)

    取 Vu=U(:,P+l:M)

    为噪声特征值对应的特征向量,即噪声子空间。

    再令 S(M-l,M-l)=0,S(M-2,M-2)=0,S(M,M)=0; (5.14)

    SS=S; RXXX=U*SS*V’ (5.15)

    低秩逼近法,用一个低秩矩阵来代替满秩矩阵RXX。

    再对RXXX进行分解

    [UU,SSS,VV]=svd(RXXX) (5.16)

    Vuu=UU(:,P+l:M)

    噪声特征值对应的特征向量,即噪声子空间。再对两次得到的噪声子空间向量进行平均 ,得到

    VU=(Vuu+Vu)./2; (5.17)

    此即为经过处理后的噪声子空间。再用这个噪声特征向量代入去计算,MUSIC算法就能有效地估计出信号的DOA了。

    这里的推导主要从 的数学特征上入手,使噪声子空间经过处理后能够与方向矢量充分正交,从而估计出信号的DOA。下面的MATLAB 仿真显示了该修正算法的有效性。

    5.3.2 修正MUSIC算法的实现

    1)  考虑从 入射的三个信号源,其中后面两个信号源是相干信号。

    图5.5 相干源的MUSIC算法与修正MUSIC算法谱图

    图中,蓝线表示MUSIC估计,红线表示修正MUSIC估计。由上图可知:修正MUSIC算法可以清晰地估计出相干信号的DOA,而传统的算法却不能估计出这类信号的DOA。

    2)从 入射的三个独立信号源,SNR分别是3dB,3dB,3dB。

    图5.6 相隔较近的小信噪比信号的MUSIC算法与修正MUSIC算法谱图

    图中,蓝线表示MUSIC算法估计,红线表示修正MUSIC算法估计。由上图可以看到,修正MUSIC算法可清晰地分辨出相隔比较近的小信噪比信号的DOA。

    3)从 入射的三个信号源,SNR分别为3dB,5dB,3dB。

    图5.7 MUSIC算法与修正MUSIC算法的性能比较

    从上面的仿真图中可以看出:修正MUSIC算法与传统的MUSIC算法对信号DOA估计出来的谱图相比较,有更高的谱峰,从而更有利于目标的分辨,特别是对于小信噪比信号。

    此外,利用MUSIC算法进行DOA估计时,随着信号信噪比的提高,对于相隔比较近的来波信号的估计有明显的改善。

    5.3.3 修正MUSIC算法的性能分析

    由以上讨论可知,修正MUSIC算法与普通MUSIC算法形式完全一样,都是利用空间谱函数

    来对信号进行DOA估计地,只是修正MUSIC算法对用于求特征分解的协方差矩阵 先进行处理,对其进行低秩逼近,使协方差矩阵满秩,以使信号子空间不渗透到噪声子空间去,从而使它们能够充分正交,达到对信号DOA进行有效估计。这样其实等效于提高了信噪比,同时也对特征根 重新进行了排列,因而有助于减小DOA估计的方差,因为

    DOA估计的方差与特征值间的关系

    (5.18)

    式中M为天线阵元数,P为入射信号数目,N为采样数, 是与特征根 MUSIC算法对信号DOA的应用

    波达方向(DOA)估计的基本问题就是确定同时处在空间某一区域内多个感兴趣的信号的空间位置(即多个信号到达阵列参考阵元的方向角)。最早的也是最经典的超分辨DOA估计方法是著名的MUSIC方法,MUSIC是多重信号分类(Multiple Signal Classification)的英文缩写。它是由R.O. Schmidt于1979年提出来的,由1986年重新发表的。MUSIC算法利用了信号子空间和噪声子空间的正交性,构造空间谱函数,通过谱峰搜索,检测信号的DOA.它是建立在以下假设基础上的:

    (1) 阵列形式为线性均匀阵,阵元间距不大于处理最高频率信号波长的二分之一;

    (2) 处理器的噪声为加性高斯分布,不同阵元间距噪声均为平稳随机过程,独立同分布,空间平稳(各阵元噪声方差相等);

    (3) 空间信号为零均值平稳随机过程,它与阵元噪声相互独立;

    (4) 信号源数小于阵列元数,信号取样数大于阵列元数,信号源为窄带信号,即信号通过天线阵的时间远远小于信号带宽的倒数.

    5.2.1 MUSIC算法的基本原理

    图5.1 均匀天线阵列

    如图5.1,M个天线阵元均匀直线排列,单元间距d为1/2个波长,布置成一个阵列天线。设有P(P

    X(n)=AS(n)+U(n)   n=1,2,……N    (5.1)

    式中X(n)= 为M个阵元输出;

    A= ,

    式中  ,T表示转置, 为载波波长,i=1,2,……,P; 为第i个平面波的复振幅;U(n)= , 为零均值、方差为 的白噪声,且与信号源不相关;N为采样数。

    信号和噪声的协方差矩阵分别为

    S= U=

    接收信号的协方差(阵列输出信号协方差)

    ,以上式中H为共轭转置   (5.2)

    因为 为MXM矩阵,所以能分解为M个特征值和特征向量,把这些特征值和特征向量用 , (i=l,2,…,M)来表示,则 可表示为

    (5.3)

    这里,V是以 为元素的列矩阵, 是以 为元素的对角矩阵。从这个分析结果,有下面重要性质:

    [性质1] 各到达波是非相干(信号间相关系数不到l),设各信号和噪声不相关,在 的特征值里,下面关系成立

    (5.4)

    即主要的特征值(信号特征值)个数和到达波束P相等,剩下的特征值(噪声特征值)的大小等于噪声功率。根据这个性质可以估计到达波的个数。进一步,按照特征值分布, 可分为信号功率和噪声功率之和

    = =  (5.5)

    V=[ ]=

    由于特征向量相互正交,则由下面第二个重要的性质。

    [性质2] 对应噪声特征值的特征向量(噪声特征向量)和各到达波的信号向量(信号特征向量)正交 。

    …M, i=1,…P.    (5.6)

    于是,阵列的空间谱函数可表示为

    (5.7)

    式中分母是信号向量和噪声向量的内积。在性质2成立时的 分母是零, 有一尖峰。MUSIC算法就是通过寻找波峰来估计到达角的。通常把信号特征矢量覆盖的空间称为信号子空间(Signal Subspace),噪声特征向量覆盖的空间称为噪声子空间(Noise Subspace)。把基于这个原理的估计到达波方向的方法称为部分空间法(Subspace Method)。MUSIC算法就是用信号或噪声子空间进行低秩信息的提取。

    5.2.2 MUSIC算法的实现

    MUSIC算法的实现步骤:

    1) 根据N个接收信号矢量得到阵列输出向量的协方差矩阵

    (5.8)

    对上面的协方差矩阵进行特征值分解

    (5.9)

    2) 然后按特征值的大小顺序,把与信号个数P相等的特征值和对应的特征向量看作信号子空间,把剩下的(M-P)个特征值和特征向量看作噪声部分空间。

    = (5.10)

    3) 使 变化,按照空间谱 来计算谱函数,通过寻找峰值来得到信号到达方向的估计值。

    以下给出基于MATLAB的MUSIC算法估计仿真:

    (1)从 入射的三个独立信号源,SNR分别为12dB,10dB,9dB。

    图5.2 MUSIC算法的谱图

    从谱图可以看出:在满足上面的假设前提下,MUSIC算法可精确估计出信号的DOA。

    尽管MUSIC算法在满足上述假设前提下可以精确估计信号的DOA,但它也有局限性:就是在低SNR和小样本的条件下无法分辨出空间相距比较近的信号。还有就是在现实当中,由于多径效应,接收到的信号一般是高相关信号,甚至是相关信号。当阵列接收到的是相干信号时,MUSIC算法就失去了其有效性,不再能估计出信号的DOA了 。

    (2)如下图,从 入射的三个信号源,SNR分别是20dB、10dB、12dB,其中,后面两个是相干信号。

    图5.3 相干源的MUSIC谱图

    由上面的谱图可以看出:MUSIC算法无法分辨出 信号,即MUSIC算法对于相干信号的DOA估计完全失效。

    (3)如下图,三个分别从 入射的信号源,SNR分别为8dB,6dB,5dB。

    图5.4 相隔比较近的小信噪比信号的MUSIC谱图

    由谱图可以看出:MUSIC算法无法分辨 和 这两个信号,即MUSIC算法对于相隔比较近的小信噪比信号的DOA估计已经失效。

    针对上述情况,就必须找到一种新的算法或对MUSIC算法进行改进,使它在能区分一般环境下信号的基础上,也能分辨出相干信号的DOA和相隔比较近的小信号比信号的DOA。下面讨论一种修正的MUSIC算法。

    5.3修正MUSIC算法对信号DOA的估计

    MUSIC算法实现对信号源DOA的估计,是基于对阵列输出信号协方差进行特征分解来估计来波方向的。然而,若信号源中有某些源是相关或完全相关(相干),相干的几个信号就可能合并成一个信号,到达阵列的独立源数将减少,即阵列输出信号协方差的秩rank( )<P,对信号协方差矩阵进行特征值分解后,某些相干源的方向矢量不正交与噪声子空间,不出现信号零点。所以,有些源在空间谱曲线中将不呈现峰值,造成谱估计的漏报。

    对于小信噪比以及角度相隔比较近的信号,它们的阵列信号协方差矩阵进行特征值分解后同样会出现类似的情况,从而不能准确地估计信号的DOA。

    因此要对MUSIC算法进行改进,就是要对阵列输出信号协方差矩阵进行处理,使信号协方差的秩恢复为rank( )=P,从而能有效地估计出信号的DOA。

    空间平滑法较好地解决了相干信号源的情况,但它是以牺牲天线的有效阵元数为条件的,同时也增加了计算量。同时它对小信噪比信号和到达角度相隔比较近的信号不能分辨。本节研究的是一种修正的MUSIC算法,它在实现MUSIC算法功能的基础上,能分辨出上述三种环境下的信号。

    5.3.1 修正MUSIC算法的基本原理

    阵列输出信号的协方差矩阵为

    其中   X(n)=AS(n)+U(n) n=1,2,……N       (5.11)

    式中X(n)= 为M个阵元输出;A= ,        , ,T表转置, 为载波波长,i=1,2,…,P; 为第i个平面波的复振幅;U(n)= , 为零均值、方差为 的白噪声,且与信号源不相关;N为采样数。

    令I为MxM反向单位矩阵,即

    I=

    构造 RXX     5.12)

    这样做是使RXX成为Hermite的Toeplitz矩阵。Toeplitz矩阵的任何一条对角线取相同元素,关于副对角线对称的。由于协方差矩阵 是Hermite的Toeplitz矩阵,所以满足 = 。阵列输出矢量N次采样数据组成矩阵X=[X( ),X( ),…X( )],协方差矩阵的估值为 。一般情况下 只是Hermite矩阵,不是Toeplitz矩阵,利用 是Toeplitz性质对 进行修正,得到Toeplitz的协方差矩阵的估值RXX= + ,显然RXX是Hermite的Toeplitz矩阵,由此可知,RXX是 的无偏估计。再对RXX进行奇异值分解 ,有

    [U,S,V]=svd(RXX) (5.13)

    取 Vu=U(:,P+l:M)

    为噪声特征值对应的特征向量,即噪声子空间。

    再令 S(M-l,M-l)=0,S(M-2,M-2)=0,S(M,M)=0; (5.14)

    SS=S; RXXX=U*SS*V’ (5.15)

    低秩逼近法,用一个低秩矩阵来代替满秩矩阵RXX。

    再对RXXX进行分解

    [UU,SSS,VV]=svd(RXXX) (5.16)

    Vuu=UU(:,P+l:M)

    噪声特征值对应的特征向量,即噪声子空间。再对两次得到的噪声子空间向量进行平均 ,得到

    VU=(Vuu+Vu)./2; (5.17)

    此即为经过处理后的噪声子空间。再用这个噪声特征向量代入去计算,MUSIC算法就能有效地估计出信号的DOA了。

    这里的推导主要从 的数学特征上入手,使噪声子空间经过处理后能够与方向矢量充分正交,从而估计出信号的DOA。下面的MATLAB 仿真显示了该修正算法的有效性。

    5.3.2 修正MUSIC算法的实现

    1)  考虑从 入射的三个信号源,其中后面两个信号源是相干信号。

    图5.5 相干源的MUSIC算法与修正MUSIC算法谱图

    图中,蓝线表示MUSIC估计,红线表示修正MUSIC估计。由上图可知:修正MUSIC算法可以清晰地估计出相干信号的DOA,而传统的算法却不能估计出这类信号的DOA。

    2)从 入射的三个独立信号源,SNR分别是3dB,3dB,3dB。

    图5.6 相隔较近的小信噪比信号的MUSIC算法与修正MUSIC算法谱图

    图中,蓝线表示MUSIC算法估计,红线表示修正MUSIC算法估计。由上图可以看到,修正MUSIC算法可清晰地分辨出相隔比较近的小信噪比信号的DOA。

    3)从 入射的三个信号源,SNR分别为3dB,5dB,3dB。

    图5.7 MUSIC算法与修正MUSIC算法的性能比较

    从上面的仿真图中可以看出:修正MUSIC算法与传统的MUSIC算法对信号DOA估计出来的谱图相比较,有更高的谱峰,从而更有利于目标的分辨,特别是对于小信噪比信号。

    此外,利用MUSIC算法进行DOA估计时,随着信号信噪比的提高,对于相隔比较近的来波信号的估计有明显的改善。

    5.3.3 修正MUSIC算法的性能分析

    由以上讨论可知,修正MUSIC算法与普通MUSIC算法形式完全一样,都是利用空间谱函数

    来对信号进行DOA估计地,只是修正MUSIC算法对用于求特征分解的协方差矩阵 先进行处理,对其进行低秩逼近,使协方差矩阵满秩,以使信号子空间不渗透到噪声子空间去,从而使它们能够充分正交,达到对信号DOA进行有效估计。这样其实等效于提高了信噪比,同时也对特征根 重新进行了排列,因而有助于减小DOA估计的方差,因为

    DOA估计的方差与特征值间的关系

    (5.18)

    式中M为天线阵元数,P为入射信号数目,N为采样数, 是与特征根 MUSIC算法对信号DOA的应用

    波达方向(DOA)估计的基本问题就是确定同时处在空间某一区域内多个感兴趣的信号的空间位置(即多个信号到达阵列参考阵元的方向角)。最早的也是最经典的超分辨DOA估计方法是著名的MUSIC方法,MUSIC是多重信号分类(Multiple Signal Classification)的英文缩写。它是由R.O. Schmidt于1979年提出来的,由1986年重新发表的。MUSIC算法利用了信号子空间和噪声子空间的正交性,构造空间谱函数,通过谱峰搜索,检测信号的DOA.它是建立在以下假设基础上的:

    (1) 阵列形式为线性均匀阵,阵元间距不大于处理最高频率信号波长的二分之一;

    (2) 处理器的噪声为加性高斯分布,不同阵元间距噪声均为平稳随机过程,独立同分布,空间平稳(各阵元噪声方差相等);

    (3) 空间信号为零均值平稳随机过程,它与阵元噪声相互独立;

    (4) 信号源数小于阵列元数,信号取样数大于阵列元数,信号源为窄带信号,即信号通过天线阵的时间远远小于信号带宽的倒数.

    5.2.1 MUSIC算法的基本原理

    图5.1 均匀天线阵列

    如图5.1,M个天线阵元均匀直线排列,单元间距d为1/2个波长,布置成一个阵列天线。设有P(P

    X(n)=AS(n)+U(n)   n=1,2,……N    (5.1)

    式中X(n)= 为M个阵元输出;

    A= ,

    式中  ,T表示转置, 为载波波长,i=1,2,……,P; 为第i个平面波的复振幅;U(n)= , 为零均值、方差为 的白噪声,且与信号源不相关;N为采样数。

    信号和噪声的协方差矩阵分别为

    S= U=

    接收信号的协方差(阵列输出信号协方差)

    ,以上式中H为共轭转置   (5.2)

    因为 为MXM矩阵,所以能分解为M个特征值和特征向量,把这些特征值和特征向量用 , (i=l,2,…,M)来表示,则 可表示为

    (5.3)

    这里,V是以 为元素的列矩阵, 是以 为元素的对角矩阵。从这个分析结果,有下面重要性质:

    [性质1] 各到达波是非相干(信号间相关系数不到l),设各信号和噪声不相关,在 的特征值里,下面关系成立

    (5.4)

    即主要的特征值(信号特征值)个数和到达波束P相等,剩下的特征值(噪声特征值)的大小等于噪声功率。根据这个性质可以估计到达波的个数。进一步,按照特征值分布, 可分为信号功率和噪声功率之和

    = =  (5.5)

    V=[ ]=

    由于特征向量相互正交,则由下面第二个重要的性质。

    [性质2] 对应噪声特征值的特征向量(噪声特征向量)和各到达波的信号向量(信号特征向量)正交 。

    …M, i=1,…P.    (5.6)

    于是,阵列的空间谱函数可表示为

    (5.7)

    式中分母是信号向量和噪声向量的内积。在性质2成立时的 分母是零, 有一尖峰。MUSIC算法就是通过寻找波峰来估计到达角的。通常把信号特征矢量覆盖的空间称为信号子空间(Signal Subspace),噪声特征向量覆盖的空间称为噪声子空间(Noise Subspace)。把基于这个原理的估计到达波方向的方法称为部分空间法(Subspace Method)。MUSIC算法就是用信号或噪声子空间进行低秩信息的提取。

    5.2.2 MUSIC算法的实现

    MUSIC算法的实现步骤:

    1) 根据N个接收信号矢量得到阵列输出向量的协方差矩阵

    (5.8)

    对上面的协方差矩阵进行特征值分解

    (5.9)

    2) 然后按特征值的大小顺序,把与信号个数P相等的特征值和对应的特征向量看作信号子空间,把剩下的(M-P)个特征值和特征向量看作噪声部分空间。

    = (5.10)

    3) 使 变化,按照空间谱 来计算谱函数,通过寻找峰值来得到信号到达方向的估计值。

    以下给出基于MATLAB的MUSIC算法估计仿真:

    (1)从 入射的三个独立信号源,SNR分别为12dB,10dB,9dB。

    图5.2 MUSIC算法的谱图

    从谱图可以看出:在满足上面的假设前提下,MUSIC算法可精确估计出信号的DOA。

    尽管MUSIC算法在满足上述假设前提下可以精确估计信号的DOA,但它也有局限性:就是在低SNR和小样本的条件下无法分辨出空间相距比较近的信号。还有就是在现实当中,由于多径效应,接收到的信号一般是高相关信号,甚至是相关信号。当阵列接收到的是相干信号时,MUSIC算法就失去了其有效性,不再能估计出信号的DOA了 。

    (2)如下图,从 入射的三个信号源,SNR分别是20dB、10dB、12dB,其中,后面两个是相干信号。

    图5.3 相干源的MUSIC谱图

    由上面的谱图可以看出:MUSIC算法无法分辨出 信号,即MUSIC算法对于相干信号的DOA估计完全失效。

    (3)如下图,三个分别从 入射的信号源,SNR分别为8dB,6dB,5dB。

    图5.4 相隔比较近的小信噪比信号的MUSIC谱图

    由谱图可以看出:MUSIC算法无法分辨 和 这两个信号,即MUSIC算法对于相隔比较近的小信噪比信号的DOA估计已经失效。

    针对上述情况,就必须找到一种新的算法或对MUSIC算法进行改进,使它在能区分一般环境下信号的基础上,也能分辨出相干信号的DOA和相隔比较近的小信号比信号的DOA。下面讨论一种修正的MUSIC算法。

    5.3修正MUSIC算法对信号DOA的估计

    MUSIC算法实现对信号源DOA的估计,是基于对阵列输出信号协方差进行特征分解来估计来波方向的。然而,若信号源中有某些源是相关或完全相关(相干),相干的几个信号就可能合并成一个信号,到达阵列的独立源数将减少,即阵列输出信号协方差的秩rank( )<P,对信号协方差矩阵进行特征值分解后,某些相干源的方向矢量不正交与噪声子空间,不出现信号零点。所以,有些源在空间谱曲线中将不呈现峰值,造成谱估计的漏报。

    对于小信噪比以及角度相隔比较近的信号,它们的阵列信号协方差矩阵进行特征值分解后同样会出现类似的情况,从而不能准确地估计信号的DOA。

    因此要对MUSIC算法进行改进,就是要对阵列输出信号协方差矩阵进行处理,使信号协方差的秩恢复为rank( )=P,从而能有效地估计出信号的DOA。

    空间平滑法较好地解决了相干信号源的情况,但它是以牺牲天线的有效阵元数为条件的,同时也增加了计算量。同时它对小信噪比信号和到达角度相隔比较近的信号不能分辨。本节研究的是一种修正的MUSIC算法,它在实现MUSIC算法功能的基础上,能分辨出上述三种环境下的信号。

    5.3.1 修正MUSIC算法的基本原理

    阵列输出信号的协方差矩阵为

    其中   X(n)=AS(n)+U(n) n=1,2,……N       (5.11)

    式中X(n)= 为M个阵元输出;A= ,        , ,T表转置, 为载波波长,i=1,2,…,P; 为第i个平面波的复振幅;U(n)= , 为零均值、方差为 的白噪声,且与信号源不相关;N为采样数。

    令I为MxM反向单位矩阵,即

    I=

    构造 RXX     5.12)

    这样做是使RXX成为Hermite的Toeplitz矩阵。Toeplitz矩阵的任何一条对角线取相同元素,关于副对角线对称的。由于协方差矩阵 是Hermite的Toeplitz矩阵,所以满足 = 。阵列输出矢量N次采样数据组成矩阵X=[X( ),X( ),…X( )],协方差矩阵的估值为 。一般情况下 只是Hermite矩阵,不是Toeplitz矩阵,利用 是Toeplitz性质对 进行修正,得到Toeplitz的协方差矩阵的估值RXX= + ,显然RXX是Hermite的Toeplitz矩阵,由此可知,RXX是 的无偏估计。再对RXX进行奇异值分解 ,有

    [U,S,V]=svd(RXX) (5.13)

    取 Vu=U(:,P+l:M)

    为噪声特征值对应的特征向量,即噪声子空间。

    再令 S(M-l,M-l)=0,S(M-2,M-2)=0,S(M,M)=0; (5.14)

    SS=S; RXXX=U*SS*V’ (5.15)

    低秩逼近法,用一个低秩矩阵来代替满秩矩阵RXX。

    再对RXXX进行分解

    [UU,SSS,VV]=svd(RXXX) (5.16)

    Vuu=UU(:,P+l:M)

    噪声特征值对应的特征向量,即噪声子空间。再对两次得到的噪声子空间向量进行平均 ,得到

    VU=(Vuu+Vu)./2; (5.17)

    此即为经过处理后的噪声子空间。再用这个噪声特征向量代入去计算,MUSIC算法就能有效地估计出信号的DOA了。

    这里的推导主要从 的数学特征上入手,使噪声子空间经过处理后能够与方向矢量充分正交,从而估计出信号的DOA。下面的MATLAB 仿真显示了该修正算法的有效性。

    5.3.2 修正MUSIC算法的实现

    1)  考虑从 入射的三个信号源,其中后面两个信号源是相干信号。

    图5.5 相干源的MUSIC算法与修正MUSIC算法谱图

    图中,蓝线表示MUSIC估计,红线表示修正MUSIC估计。由上图可知:修正MUSIC算法可以清晰地估计出相干信号的DOA,而传统的算法却不能估计出这类信号的DOA。

    2)从 入射的三个独立信号源,SNR分别是3dB,3dB,3dB。

    图5.6 相隔较近的小信噪比信号的MUSIC算法与修正MUSIC算法谱图

    图中,蓝线表示MUSIC算法估计,红线表示修正MUSIC算法估计。由上图可以看到,修正MUSIC算法可清晰地分辨出相隔比较近的小信噪比信号的DOA。

    3)从 入射的三个信号源,SNR分别为3dB,5dB,3dB。

    图5.7 MUSIC算法与修正MUSIC算法的性能比较

    从上面的仿真图中可以看出:修正MUSIC算法与传统的MUSIC算法对信号DOA估计出来的谱图相比较,有更高的谱峰,从而更有利于目标的分辨,特别是对于小信噪比信号。

    此外,利用MUSIC算法进行DOA估计时,随着信号信噪比的提高,对于相隔比较近的来波信号的估计有明显的改善。

    5.3.3 修正MUSIC算法的性能分析

    由以上讨论可知,修正MUSIC算法与普通MUSIC算法形式完全一样,都是利用空间谱函数

    来对信号进行DOA估计地,只是修正MUSIC算法对用于求特征分解的协方差矩阵 先进行处理,对其进行低秩逼近,使协方差矩阵满秩,以使信号子空间不渗透到噪声子空间去,从而使它们能够充分正交,达到对信号DOA进行有效估计。这样其实等效于提高了信噪比,同时也对特征根 重新进行了排列,因而有助于减小DOA估计的方差,因为

    DOA估计的方差与特征值间的关系

    (5.18)

    式中M为天线阵元数,P为入射信号数目,N为采样数, 是与特征根 对应的特征向量, 为噪声方差, 为信号入射实际角度, 为估计角度;

    (5.19)

    式中 其中

    由上面DOA估计的方差公式我们可以看出,信号子空间特征矢量所对应的特征值 (m=l,2,…P)越接近于噪声方差 ,即 的值越小,则 的值就越大,即DOA估计的方差也越大,算法的性能较差;反之,如果 的值越大,则 的值就越小,算法性能就越小。

    展开全文
  • E-mail:chengwenchi1986@gmail.com摘要:本文主要是对 DOA(波达方向)估计中传统 MUSIC算法及其改进算法作了简要的介绍,主要包括了MUSIC算法,求根MUSIC算法,循环MUSIC算法,波束空间MUSIC算法,SMARTMUSIC算法。...

    E-mail:chengwenchi1986@gmail.com

    摘要:本文主要是对 DOA(波达方向)估计中传统 MUSIC

    算法及其改进算法作了简要

    的介绍,主要包括了MUSIC算法,求根MUSIC算法,循环MUSIC算法,波束空间MUSIC算法,SMART

    MUSIC算法。并且在对每个算法做了原理性的分析的基础上给出了简要的性能分析。

    关键词:DOA估计;MUSIC

    算法;求根MUSIC算法;循环MUSIC算法

    Abstract:The paper has a brief introduction on the traditional

    MUSIC algorithm and the modified of the DOA estimation.It is mainly

    about the extract MUSIC algorithm、circle MUSIC algorithm、beam

    space MUSIC algorithm and SMART MUSIC algorithm.It also give some

    brief behaviour analysis after analyzing each algorithm’s

    principle.

    Key words:DOA estimation; MUSIC algorithm; extract MUSIC

    algorithm; circle MUSIC algorithm

    1. 引言

    波达方向(Direction Of

    Arrival,DOA)技术,就是根据来波信号估计其方位角的信号处理技术.为了降低移动通信系统中的多址干扰、降低发射功率和提高系统容量,智能天线目前成为研究的热点.它引入了空分多址的概念,通过用户空间位置的差异对其进行分离.因此各用户的DOA作为反映用户空间位置的重要参量在智能天线中扮演着非常重要的角色。如何准确的估计各个用户DOA是非常值得研究的领域。

    DOA估计的研究工作大约从20世纪60年代开始,在产生的多种算法中,尤其Schmidt的MUSIC(Multiple

    Signal

    Classification)法最为著名,这种方法获得了广泛的应用。虽然MUSIC算法有着非常优异的性能,然而其计算复杂度和对系统存储的巨大需求使得其在应用中遇到不少困难。于是在原来MUSIC的基础上又诞生了求根MUSIC算法、约束MUSIC算法、波束空间MUSIC算法等。

    2 . 各算法分析及性能介绍

    2.1 MUSIC算法之前的DOA估计算法

    DOA估计的传统方法主要基于波束形成和零陷引导的概念,并没有利用到接受信号矢量的模型或者是信号和噪声的统计模型。阵列流形知道以后,传统的DOA估计方法就可以利用波束形成技术把波束调节到任意方向,寻找出输出功率的峰值。传统的DOA方法有延迟——相加法和Capon最小方差方法等。

    2.2 MUSIC算法

    2.2.1MUSIC算法原理:

    MUSIC是多重信号分类(Multiple Signal

    Classification)的英文缩写。它是由R.O.

    Schmidt于1979年提出来的,由1986年重新发表的。MUSIC算法利用了信号子空间和噪声子空间的正交性,构造空间谱函数,通过谱峰搜索,检测信号的DOA。它是建立在以下假设基础上的:

    1, 阵列形式为线性均匀阵,阵元间距不大于处理最高频率信号波长的二分之一;

    2, 处理器的噪声为加性高斯分布,不同阵元间距噪声均为平稳随机过程,且相互独立,空间平稳(各阵元噪声方差相等);

    3, 空间信号为零均值平稳随机过程,它与阵元噪声相互独立;

    4, 信号源数小于阵列元数,信号取样数大于阵列元数。

    在此假设基础上,MUSIC算法对波达方向DOA的估计理论上可以有任意高的分辨率。

    MUSIC算法是一种基于特征结构的高分辨率DOA算法,这一类方法都是基于接受信号相关矩阵的以下两个特性:1,相关矩阵的特征向量张成的矢量空间可以分成两个部分,信号子空间和噪声子空间;2,不同方向信号源对应的阵列流形矢量与噪声子空间正交。由于噪声子空间与信号子空间正交,所以这些阵列流形矢量就包含在信号子空间中。(噪声子空间是由相关矩阵的小特征值对应的特征向量所张成,而信号子空间则由相关矩阵大特征值对应的特征向量所张成。

    如图,M个天线阵元均匀直线排列,单元间距d为1/2个波长,布置成一个阵列天线。有P(P

    , , … 。在第n次采样时刻,得到的数据向量为

    其中, 是入射信号矢量; 是噪声矢量;A是阵列方向矢量矩阵,

    MUSIC算法需要满足以下前提条件:

    l M>d, 并且对应于不同的 的阵列方向矢量独立;

    l , , (这些对于噪声的假定对MUSIC

    算法来说至关重要);

    l 矩阵 是非奇异的正定矩阵;

    MUSIC算法假定输入信号与噪声互不相关,则输入信号的自相关矩阵为

    其中, 是信号自相关矩阵。对 进行特征值分解,得到M个特征值

    ,并且满足 ,利用上式进行分解,得

    显然, 的特征值是

    。若入射信号互不相关,则矩阵A列满秩,并且信号相关矩阵也满秩。

    由于矩阵A是满秩的,并且

    也是非奇异的,所以当入射信号个数小于阵列天线的阵元个数时,矩阵

    是半正定的,且秩为d.

    根据线性代数的知识,矩阵 值 当中,有M-d个为零。由式 可知,

    的特征值中有k=M-d个等于噪声的方差 。

    关于特征值 的特征向量为 ,且满足:

    对应于M-d个最小特征值的特征向量,有

    由于矩阵A是满秩的,所以矩阵 也是非奇异的,即有

    。上式表明,M-d个最小特征值对应的特征矢量与构成矩阵A的d个方向导引矢量正交,这就是MUSIC算法的核心思想。

    为了寻找出噪声子空间,需要构建一个包含噪声特征矢量的矩阵:

    因为对应于信号分量的方向导引矢量与噪声子空间特征矢量相互正交,多个入射信号的DOA估计值就可以通过确定MUSIC空间谱的峰值而做出估计,这些峰值由

    给出。 和

    的正交性使得分母达到最小值,从而得到上式定义的MUSIC谱的峰值。MUSIC谱中d个最大峰值对应于入射到阵列上的d个信号波达方向。

    MUSIC算法的基本步骤:

    1) 获得输入信号的采样值 ,k=0,…,K-1,估计输入信号的协方差矩阵:

    2) 对 进行本征值分解:

    其中,  , 为 的特征值,

    为与这些特征值对应的特征矢量构成的矩阵。

    3) 利用最小特征值 的重数K估计信号数目。

    4) 计算MUSIC谱。

    5) 找出 的 个最大峰值,得到波达方向的估计值。

    2.2.2MUSIC算法性能分析:

    MUSIC算法具有极高的空间分辨率,可以区分空间两个十分靠近的信号。和传统的DOA估计方法不同。MUSIC算法空间谱在估计信号功率时并没有考虑波达方向角。当精确知道阵列输入协方差矩阵的集平均时,在非相关的相同噪声环境下,可以确保

    的峰值对应真实的信号波达方向角。由于这些峰值是可以区分的,与波达方向角的真实间隔无关,所以理论上将,只要阵列的校准足够准确,就可以达到极高的空间分辨率。但是,MUSIC算法也存在以下缺点:

    1, 当入射信号高度相关时,信号的子相关矩阵就变为奇异了,此时MUSIC算法将会失效。

    2, MUSIC算法里做了一个假设,就是到达信号的个数是已知的,但实际中达到的信号的个数确是未知的。通过研究特征值的分布方法来估计达到信号的个数是可能的,然而特征值的估计是依赖协方差矩阵的估计值。在SNR比较低时,使用估计的接受信号的数据少时,就很难判断了。

    3,MUSIC算法有一个谱峰搜索的过程,而这个过程的计算量巨大。

    2.3求根MUSIC算法:

    2.3.1求根MUSIC算法原理

    对于阵元间距为d的等距直线阵列,导引向量

    的第m个元素可以表示为

    则MUSIC谱函数可以写成:

    其中

    是矩阵C中第L条对角线的元素之和。

    定义多项式:

    研究MUSIC谱的特性也就可以通过研究多项式

    来完成。在没有噪声的理想情况下,极点落在单位圆上,位置由波达方向决定,即

    处产生MUSIC谱的一个波蜂,此时

    2.3.2求根MUSIC算法性能分析:

    MUSIC算法进行DOA估计时需要进行空间谱峰搜索,计算量巨大。当天线阵列是均匀线性阵列的时候,求根MUSIC算法具有极佳的性能和计算效率。

    因为把计算量巨大的谱峰搜索转换为对多项式

    的研究,大大节省了计算量,而且Barbell通过计算机仿真证明,求根MUSIC比MUSIC谱形式具有更佳的分辨率,特别是在低SNR的情况下。但是,由于该算法是建立在天线阵列是均匀线性阵列的基础上的,应用面比较窄。此算法对于均匀圆阵也适用。

    2.4循环MUSIC算法:

    2.4.1循环MUSIC算法原理:

    研究一个有M个阵元的天线阵列,假设接受的d个信号在频率

    处具有谱相干性,并且干扰信号在这个频率上没有相干性。这种情况在严重的共信道干扰环境中检测具有特性谱相关和多径分量的信号时经常会遇到。

    令 ,i=0,1,…,d-1为目标信号,

    为干扰和噪声矢量,则接受信号矢量可以表示为

    因为目标信号为频率 具有谱相关性,接受信号的循环自相关矩阵

    定义为

    其中, 是目标信号的循环自相关矩阵,定义为

    其中,符号“ ”表示 。

    显然, 的秩为d,对于d

    的零空间又对应于零特征值的特征矢量:

    如果信号不完全相关,则

    满秩,为d。由于A也是满秩矩阵,有上面的推导可知,

    的零空间正交与目标信号的方向矢量,即

    ,i=0,1,…,

    将上式作为正交性的量度,则循环MUSIC谱可以定义为

    对于所有可能的

    值进行谱峰搜索,就可以得到目标信号的DOA估计值。

    2.4.2循环MUSIC算法性能分析:

    循环MUSIC算法是利用接受信号的谱相干性和空间相干性的DOA估计方法。将谱的相干性和MUSIC算法结合起来,在相距很近的信号中只有一个期望信号并且信号的间隔小于阵列的分辨率时,也能够分辨出这个有用信号。而且,循环MUSIC算法不受入射信号个数要求的限制。

    2.5波束空间MUSIC算法:

    2.5.1波束空间MUSIC算法原理:

    假设有D个窄带信号入射到一个由N个阵元构成的天线阵列上,同时,假定入射信号之间信号互不相关。入射信号表示为

    ,i=1,2,…,D

    第i个信号的阵列方向矢量表示为 ,其中 是信号的入射角。

    是一个N维的列矢量,每一个元素表示了天线阵元信号相对于参考阵元的相对位移,由信号的入射角度和天线阵元的空间位置共同决定。对于均匀线性直线阵列而言,有

    其中,

    和d分别是信号波长和阵元间距。定义信号矢量和阵列方向矩阵为

    定义第k个阵元上接受信号为

    ,k=1,2,…,N。接受信号中除了入射信号以外还有信道、发射机、接收机等产生的噪声。定义阵列输入(接受)信号矢量为

    考虑到窄带信号的假定,则输入信号矢量可以用矩阵形式表示为

    其中, 是N维的噪声矢量,一般是均值为0、方差(功率)为

    的复高斯随机工程,并且与各个入射信号统计独立。

    在波束空间MUSIC算法中,阵列输出矢量

    在进行MUSIC算法之前要通过一个波束形成器得到输出矢量

    其中, 是

    维的波束形成矩阵,包含着方向信息的矢量,可以产生若干个对准合适方向的波束。通过上面的变换将一个N维的阵元空间列矢量变换为M维的波束空间的列矢量。同时假定矩阵

    是正交的,即满足 。

    当 时,就是传统的信号空间MUSIC算法。

    2.5.2波束空间MUSIC算法性能分析:

    与阵元空间DOA算法相比,波束空间DOA算法具有许多优势:减少计算量,提高分辨率,减少由于系统误差造成的估计误差,减少估计偏差等。

    由于波束空间MUSIC算法使用了波束形成器,波束数目少于天线阵元数目,因而算法处理的数据要比阵元空间MUSIC少,从而提高了算法的效率。

    从阵列天线的自由度的角度看,一个天线阵列的自由度等于它的阵元数目减1,而在波束空间算法中,其自由度等于波束数目。因此,实际上波束空间算法降低了天线的自由度。一般而言,需要M+1个自由度就可以对M个信源的DOA进行估计。

    2.6 SMART MUSIC算法:

    2.6 .1SMART MUSIC算法原理:

    考虑一个M元均匀线性天线阵列,阵元为全向天线,阵元间距d。有P(P

    ( ),入射角 ( )。空间噪声

    为各态遍历高斯噪声,均值为0。假定入射信号为窄带信号,波长为

    ,则M维接受信号矢量可以表示为

    其中 是阵列方向向量:

    从向量 中抽出一个L维的子向量 ( ),有

    当满足 时,

    当满足 时,

    可以证明,向量 的子向量的相关矩阵C满足 .

    最后用

    来估计DOA的值。

    其中,  ,

    2.6.2SMART MUSIC算法性能分析:

    在众多的DOA估计算法中,采用前向、后向空间平滑技术的MUSIC算法被认为是性能出众的一种。然而由于该算法需要对于接受信号相关矩阵进行本征值分析和MUSIC本征谱估计,因而计算量巨大,在需要实时处理的领域应用困难。但是SMART

    MUSIC算法在具有相同分辨率的情况下

    计算量较小,比较容易应用于实践。

    3.结论

    本文从各种基于MUSIC算法的改进算法的原理入手,从理论角度分析了各算法的推导过程,并在每节最后给出了简要的性能分析。通过分析我们可以看到,在对普通环境下信号DOA估计时,与传统MUSIC算法相比较,其它改进的MUSIC算法估计出来的谱图有更高的谱峰,更容易区别出信号的DOA,从而更有利于信号的分辨。对于相干信号,传统MUSIC算法已经不能区分它们的DOA了,而一些改进的MUSIC算法则能有效地区分出它们的DOA。但是我们也应该看到,各种改进的MUSIC算法都有自己一定的局限性,在对于不同的情况下,我们应该酌情考虑,选择合适的DOA估计方法应用。

    4.参考文献

    [1] 高星辉

    ,华南师范大学,《修正MUSIC算法对信号DOA的估计》,

    “中国优秀硕士学位论文全文数据库”

    2002年

    [2]

    韩卫杰,西南交通大学,《改进MUSIC算法在DOA估计中的研究》,“中国优秀硕士学位论文全文数据库”

    2006年

    [3]金荣洪 耿军平范瑜

    ,《无线通信中的智能天线》,北京邮电大学出版社 2006年

    [4] R. O .Schmidt,Multiple Emitter Location

    and Signal Parameter Estimation, In Proc. RADC

    Spectrum Estimation Workshop, Oct 1979,or IEEE. Trams, AP-34, No3, pp276-280, Mar, 1986

    展开全文
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